Appalachian set theory

January 27, 2007

University of North Carolina at Charlotte

Stevo Todorcevic : "Coherent sequences"



Stevo Todorcevic, "Coherent sequences", chapter to appear in the Handbook of Set Theory (PDF)

Charlotte lodging information

Post-workshop materials:

Lecture notes from this workshop by Roberto Pichardo Mendoza (PDF)

    I intend to cover first of all some basics about walks and associated statistics such as upper and lower traces, the distance functions associated to the number of steps and maximal weights. As we go on, time permitting, I will mention some applications. Among applications, I will mention the use of the subadditive distance function ρ in constructing Banach spaces with strong conditional structure but most of the applications will be set-theoretical in character to fit the composition of the audience. For example, I will explain why every one of the standard distance function ρi leads us to a canonical uncountable linear ordering C(ρi). The canonical nature of C(ρi) is seen on one hand from the facts that its Cartesian square can be decomposed into countably many chains and from the other that under some mild assumption like MAω1, C(ρi) is a minimal uncountable linear ordering in the sense that it embeds into all of its uncountable subsets. Thus, C(ρi) must appear on any basis for uncountable linear ordering as long as one tries to be compatible with MAω1. Another main theme of the series of lecture will be the so called square-bracket operation that gives us a way of reducing quantification over unbounded subsets of a regular uncountable cardinal θ to quantifications over closed unbounded subsets of θ. This reduction of quantifiers is quite useful in constructing various mathematical structures such as bilinear forms on vector spaces (i.e., so called symplectic spaces), groups, projective geometries, Banach spaces with few operators, topological spaces with some extremal properties, etc. On a set-theoretical side I will try to explain the role of imposing coherence condition on a given C-sequence on which we base our walks. This will give us some interesting results about Jensen's square sequences and their natural generalizations. We shall also use these methods to shed light on other well-known set-theoretic themes such as Chang's conjecture, reflection of stationary sets, etc. I will try to present this assuming little or no prerequisites but of course any basic familiarity with the notions such ordinal, cardinal, stationary set, could prove useful when trying to follow my series of lectures.